A189244 The n-th derivative of e^((2-x-x^2)/(1-x-x^2)), evaluated at x=1.

1, 3, -7, 9, 177, -3897, 65649, -1057851, 16606977, -238404789, 2305262889, 33442089057, -3560906733903, 182521828278351, -8055082800686367, 338022326927690397, -13915405899740874879, 566988435851123595411, -22784764731442383689127, 888283409438427072329529

Offset

0,2

Comments

The n-th derivative of exp((2-x-x^2)/(1-x-x^2) is A(n,x) = n!*sum(m=1..n, sum(k=m..n, binomial(k-1,m-1)*binomial(k,n-k)*(2*x+1)^(2*k-n) * (-x^2-x+1)^(-m-k))/m!).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..388

Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.

Formula

a(n) = n!*sum(m=1..n, sum(k=m..n, binomial(k-1,m-1) *binomial(k,n-k) * (-1)^(m+k)*3^(2*k-n))/m!), a(0)=1.

E.g.f.: exp(x*(3+x)/(3*x+x^2+1)). - Alois P. Heinz, Sep 27 2016

Mathematica

f[x_] := E^((2 - x - x^2)/(1 - x - x^2));
a[n_] := Derivative[n][f][1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 27 2018 *)

Prog

(Maxima)
a(n):=n!*sum(sum(binomial(k-1, m-1)*binomial(k, n-k)*(-1)^(m+k) * 3^(2*k-n), k, m, n)/m!, m, 1, n)

Crossrefs

Sequence in context: A337613 A152607 A118559 * A127789 A112105 A065501

Adjacent sequences: A189241 A189242 A189243 * A189245 A189246 A189247

Keyword

sign

Author

Vladimir Kruchinin, Apr 26 2011

Status

approved